How are Differential Equations usually found in real world applications?

[DrummingAtom]

I just started learning about DE's and am impressed by how much information they contain. Are they usually just a relationship between several simpler functions?

For instance, a basic kinematic problem like v = at + v_o seems quite hard to come up with experimentally but v = d/t would be easy. Thanks for any help.

 

[galoisjr]

Technically any derivative is a differential equation, just a trivial one.
For instance
d(f(x))/dx=f'(x)
since first derivatives can be treated as fractions,
d(f(x))=f'(x)dx
then integrating we get f(x)=int(f'(x)dx)

But obviously DE's can come in all shapes and forms and there are even such things as partial differential equations, integral equations, and difference equations which are very closely related to differential equations. But yes and no, any solution of a differential equation is in terms of some function. This is because your given an equation that is in terms of a function's derivatives and your asked to find the function which satisfies that equation. As I showed before sometimes it is simple algebra tricks and then integration. But for most higher order differential equations there are various methods and some of them are very complicated, while some are brute force methods such as series solutions, while still some methods are very elegant. In physics the first higher order equations that you will come across are usually oscillation. As for your last question the first equation v=at+v(0) is actually the solution of one of those trivial differential equations that I showed you before where v(0) is the initial velocity, then for constant acceleration a(t)=d(v(t))/dt=a implies that d(v(t))=adt and integrating we obtain that v(t)=at+C where C is the constant of integration and evaluating at 0 we have v(0)=0+C=C which implies that v(t)=at+v(0). You are very right though, they are an extremely elegant method to solving physical problems.

 

[rock.freak667]

In a beam bending with a uniform load w, the displacement can be related to

EI d⁴v/dx⁴ +w = 0

and applying certain finite methods one can find the displacement (angular and linear) at some point in the beam.

 

[yus310]

heat transfer, fluid mechanics, physics, biology, chemistry, all types of applications ranging to describe heat diffusion to cell displacement on a substrate

... they are fun, powerful mathematics

 

[DrummingAtom]

I should have retitled this thread to "How are Differential Equations usually *discovered* in real world applications?"

Are basic relationships usually found first then the derivatives of those then applied to a differential equations? Or are the differential equations usually found first?

Thinking from an experimental standpoint it would seem that the basic relationships are found first then the complex part later. The reason I ask is because the examples in my book seem to suggest the D.E's are found easily. Then instead of coming up with the D.E.'s as exercises we just use them and solve them. By the way, are there any books that focus on deriving differential equations instead of just solving them? Would this be called modeling D.E.'s?